1. Consider a Dirichlet series
Exercise 1, 9/08/2005
an n s .
(a) If the series converges for s = s0 , then it converges uniformly in any angular region
P C : +
arg(s s0 )
dened by a real number such that 0 < < .
(b) According to (a)
Exercise 10, 11/20/2005
1. Let F be a number eld. Find an explicit bound, C , in terms of jdisc(F=Q)j, and rI ; rP for
F , such that every element of the ideal class group for F is represented by an ideal I & yF
such that jyF =I j C . Illustrate this meth
Exercise 9, 11/10/2005
1. Determine explicitly the Pontriagin dual of Zp and Qp .
2. Determine the Pontiagin dual of Z, where Z is the pronite completion of Z.
3. Identify C with its own Pontriagin dual via
(z; w) U3 exp 2
Determine the self-
Exercise 8, 10/25/2005
1. Let F be a global eld.
(i) Is the set
a closed subset of
= (x ) P A :
(ii) Let f (t) be a polynomial over F . Let
0 be a positive real number. Is the set
= (x ) P A :
( )j T= a
Exercise 7, 10/23/2005
1. Let G = Gal(Q(3 ; 3 2)=Q). Determine the upper and lower numbering ltration of the
decompostion subgroup of G above p = 2 and p = 3.
2. Let G = Gal Qp (p2 )=Qp , where p is an odd prime number. Determine the upper and
Exercise 6, 10/05/2005
Denition Let L/K be a nite separable extension eld. The discriminant of a K -basis
1 , . . . , n of L is dened to be
d(1 , . . . , n ) := det (i (j )2
where 1 , . . . , n are the K -embeddings of L into Lsep . If is an element of L
Exercise 5, 10/03/2005
1. Determine explicitly the left invariant Haar measure on the locally compact group GL2 (Qp ).
2. Let f (x) be the characteristic function on Zp . Consider the integral
f (x)jjxjjs dx
Determine all s P C for which the abov
Exercise 3a, 10/9/2005
Correction to Exercise 3 and discussion of related issues
1. Comments on Problem 3, Exercise set 3.
(i) Statements (i), (ii) of are wrong. In fact the abscissa of convergence for F (s) is at most
1 ; see Problem 3 below.
Exercise 3, 9/18/2005
This short set of problems provides a counter-example for Question 3 in Exercise 1.
n!1 an n be a Dirichlet series which is divergent for s = 0.
Prove that the abscissa
of convergence of this Dirichlet series is equal to
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